Actually, I've resolved how get an analytic solution. First of all, you only look at the positive $x$ side of the area, then divide it into an upper and lower portion. You should be able to do the lower portion by yourself, it a hemisphere and it volume is $4\pi r^3/6$. Now for the upper portion, as I showed for a similar post by someone else (it's a homework problem, right?), here is how to do the upper part.
Let's consider the problem of rotating the upper curve, a circular arc, about the $y$-axis. Let's just call the curve $y(x)$ for the moment. I think it's well known and understood that
The problem here is that it's not immediateley obvious how to express $y(x)$. However, if the diameter of the arc is $r$, then we can write $y(x)$ as a quarter circle centered at $(x,y)=(r,r)$ as follows
We have now parameterized $x$ and $y$, so we need to rewrite the volume expression as
You should be able to take it from here. If not I can extend this answer.